Applied Mathematics & Information Sciences

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We investigate integral Galois stable representations of finite groups over local and global fields and their integers under the ground field extensions related to permutation modules. We consider normal extensions E/F and subgroups G ⊂ GLn(E) stable under the natural operation of the Galois group of E/F with some extra integrality conditions. The possible realization fields of G with these integrality conditions are of special interest, and we consider certain criteria of integrality for representations in GLn(E). We also study a series of related arithmetic problems and examples.